1. 1 Project Scheduling Contents 1. Problem Statement 2. Critical Path Method 3. Extensions to the classical project scheduling problems Literature Operations Scheduling with Applications in Manufacturing and Services, Michael Pinedo andXiuli Chao, McGraw Hill, 2000, Chapter 4

2. 2 Environment: parallel-machines jobs are subject to precedence constraints Objective: to minimise the makespan P  | prec | C max m  nCritical Path Method Pm | prec | C max 2  m<n NP hard slack job: the start of its processing time can be postponed without increasing the makespan critical job: the job that can not be postponed critical path: the set of critical jobs Problem Statement

3. 3 Critical Path Method Forward procedure that yields a schedule with minimum makespan. Notation: p j processing time of job j S' j the earliest possible starting time of job j C' j the earliest possible completion time of job j C' j = S' j +p j { all k  j } jobs that are predecessors of job j Step 1. For each job j that has no predecessors S' j =0 and C' j =p j Step 2. Compute inductively for each remaining job j C' j = S' j +p j Step 3. C max = max(C' 1,...,C' n )

4. 4 Backward procedure determines the latest possible starting and completion times. Notation: C'' j the latest possible completion time of job j S'' j the latest possible starting time of job j { j  all k } jobs that are successors of job j Step 1. For each job j that has no successors C'' j =C max and S'' j =C max -p j Step 2. Compute inductively for each remaining job j S'' j = C' j - p j Step 3. Verify that 0 = min(S'' 1,..., S'' n )

5. 5 The jobs whose earliest possible starting times are earlier than latest possible starting times are referred to as slack jobs. The jobs whose earliest possible starting times are equal to their latest possible starting times are critical jobs. A critical path is a chain of jobs which begin at time 0 and ends at C max.

6. 6 Example 12 6 345 7 8 9 4 9 8 8 33 6 12 6

7. 7 12 6 345 7 8 9 4 9 8 8 33 6 6

8. 8 12 6 345 7 8 9 4 9 8 8 33 6 Critical Path 6

9. 9 Extensions to the classical project scheduling problems Stochastic activity (job) durations Nonavailability of resources Multiple resource types Preemption of activities Multiple projects with individual project due-dates Objectives: common one: minimising overall project duration resource leveling minimise resource loading peaks without increasing project duration maximise resource utilisation factors